When I first learned about the limit it greatly disturbed my algebraic thinking, but I pushed the thought aside for a while, and now it's come back.

Take for example:

$\displaystyle \lim_{x \to 2}\frac{x^2-4}{x^2-2x}$

We have the expression

$\displaystyle \frac{x^2-4}{x^2-2x}$, which is equivalent to $\displaystyle \frac{x+2}{x}$

On the left, substituting in x we get an undefined answer, substituting into the right expression however, we get 2.

Aren't the two expression as perfect as each other? Then why do they give a different result when you substitute in x?

It's like we are, by manipulating the number, forcing it to give a certain answers against its will.